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Properties

Label 4008.2.a
Level $4008$
Weight $2$
Character orbit 4008.a
Rep. character $\chi_{4008}(1,\cdot)$
Character field $\Q$
Dimension $82$
Newform subspaces $13$
Sturm bound $1344$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4008.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(1344\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4008))\).

Total New Old
Modular forms 680 82 598
Cusp forms 665 82 583
Eisenstein series 15 0 15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(167\)FrickeDim
\(+\)\(+\)\(+\)$+$\(10\)
\(+\)\(+\)\(-\)$-$\(11\)
\(+\)\(-\)\(+\)$-$\(14\)
\(+\)\(-\)\(-\)$+$\(7\)
\(-\)\(+\)\(+\)$-$\(8\)
\(-\)\(+\)\(-\)$+$\(11\)
\(-\)\(-\)\(+\)$+$\(9\)
\(-\)\(-\)\(-\)$-$\(12\)
Plus space\(+\)\(37\)
Minus space\(-\)\(45\)

Trace form

\( 82 q + 2 q^{3} + 4 q^{5} + 82 q^{9} + O(q^{10}) \) \( 82 q + 2 q^{3} + 4 q^{5} + 82 q^{9} + 8 q^{13} - 4 q^{15} + 4 q^{17} + 16 q^{19} + 16 q^{23} + 86 q^{25} + 2 q^{27} - 4 q^{29} - 8 q^{31} + 8 q^{33} + 24 q^{35} - 8 q^{37} + 4 q^{39} - 4 q^{41} - 4 q^{43} + 4 q^{45} - 8 q^{47} + 82 q^{49} - 8 q^{51} - 12 q^{53} + 16 q^{55} - 8 q^{57} - 8 q^{59} - 20 q^{61} - 32 q^{65} - 28 q^{67} - 8 q^{69} - 8 q^{71} + 12 q^{73} - 2 q^{75} + 24 q^{79} + 82 q^{81} + 16 q^{83} + 24 q^{85} - 4 q^{87} + 4 q^{89} + 8 q^{91} + 16 q^{93} - 32 q^{95} - 4 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 167
4008.2.a.a 4008.a 1.a $1$ $32.004$ \(\Q\) None \(0\) \(-1\) \(1\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-3q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
4008.2.a.b 4008.a 1.a $1$ $32.004$ \(\Q\) None \(0\) \(1\) \(-1\) \(-1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
4008.2.a.c 4008.a 1.a $1$ $32.004$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{9}+2q^{17}-8q^{19}-4q^{23}+\cdots\)
4008.2.a.d 4008.a 1.a $1$ $32.004$ \(\Q\) None \(0\) \(1\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
4008.2.a.e 4008.a 1.a $3$ $32.004$ 3.3.148.1 None \(0\) \(-3\) \(6\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(2-\beta _{2})q^{5}+(-2+2\beta _{1})q^{7}+\cdots\)
4008.2.a.f 4008.a 1.a $5$ $32.004$ 5.5.284897.1 None \(0\) \(5\) \(1\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-\beta _{3}+\beta _{4})q^{5}+(-1-\beta _{2}+\cdots)q^{7}+\cdots\)
4008.2.a.g 4008.a 1.a $7$ $32.004$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-7\) \(-3\) \(8\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta _{1}q^{5}+(1+\beta _{2})q^{7}+q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
4008.2.a.h 4008.a 1.a $8$ $32.004$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-8\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta _{1}q^{5}+\beta _{3}q^{7}+q^{9}+(\beta _{2}+\beta _{4}+\cdots)q^{11}+\cdots\)
4008.2.a.i 4008.a 1.a $9$ $32.004$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(9\) \(-6\) \(-11\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1+\beta _{1})q^{5}+(-2+\beta _{5}+\beta _{7}+\cdots)q^{7}+\cdots\)
4008.2.a.j 4008.a 1.a $10$ $32.004$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(-10\) \(-10\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1-\beta _{1})q^{5}+(\beta _{2}+\beta _{4}+\beta _{7}+\cdots)q^{7}+\cdots\)
4008.2.a.k 4008.a 1.a $11$ $32.004$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(-11\) \(10\) \(-1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(1-\beta _{1})q^{5}+\beta _{3}q^{7}+q^{9}-\beta _{10}q^{11}+\cdots\)
4008.2.a.l 4008.a 1.a $12$ $32.004$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(12\) \(4\) \(11\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta _{1}q^{5}+(1-\beta _{9})q^{7}+q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)
4008.2.a.m 4008.a 1.a $13$ $32.004$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(0\) \(13\) \(2\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta _{1}q^{5}-\beta _{9}q^{7}+q^{9}+(1+\beta _{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4008))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4008)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(167))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(334))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(501))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(668))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1002))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1336))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2004))\)\(^{\oplus 2}\)